The reciprocal inhibition of antagonistic muscle upon lateral gaze is explained by:

Synergistic & Antagonistic Muscles (Sherrington's Law) Synergistic muscles are those that have the same field of action. Thus, for veical gaze, the superior rectus and inferior oblique muscles are synergists in moving the eye upward. Muscles synergistic for one function may be antagonistic for another. For example, the superior rectus and inferior oblique muscles are antagonists for torsion, the superior rectus causing intorsion and the inferior oblique extorsion. The extraocular muscles, like skeletal muscles, show reciprocal innervation of antagonistic muscles (Sherrington's law). Thus, in dextroversion (right gaze), the right medial and left lateral rectus muscles are inhibited while the right lateral and left medial rectus muscles are stimulated. What is reciprocal innervation? When a stretch reflex occurs, the muscles that antagonize the action of the muscle involved (antagonists) relax. This phenomenon is said to be due to reciprocal innervation. Impulses in the Ia fibers from the muscle spindles of the protagonist muscle cause postsynaptic inhibition of the motor neurons to the antagonists. The pathway mediating this effect is bisynaptic. A collateral from each Ia fiber passes in the spinal cord to an inhibitory interneuron that synapses on a motor neuron supplying the antagonist muscles. Ref: Motley W., Asbury T. (2011). Chapter 12. Strabismus. In P. Riordan-Eva, E.T. Cunningham, Jr. (Eds), Vaughan & Asbury's General Ophthalmology, 18e.